IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v7y2019i7p600-d246062.html
   My bibliography  Save this article

A Mixed Finite Volume Element Method for Time-Fractional Reaction-Diffusion Equations on Triangular Grids

Author

Listed:
  • Jie Zhao

    (School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
    School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, China)

  • Hong Li

    (School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China)

  • Zhichao Fang

    (School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China)

  • Yang Liu

    (School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China)

Abstract

In this article, the time-fractional reaction-diffusion equations are solved by using a mixed finite volume element (MFVE) method and the L 1 -formula of approximating the Caputo fractional derivative. The existence, uniqueness and unconditional stability analysis for the fully discrete MFVE scheme are given. A priori error estimates for the scalar unknown variable (in L 2 ( Ω ) -norm) and the vector-valued auxiliary variable (in ( L 2 ( Ω ) ) 2 -norm and H ( div , Ω ) -norm) are derived. Finally, two numerical examples in one-dimensional and two-dimensional spatial regions are given to examine the feasibility and effectiveness.

Suggested Citation

  • Jie Zhao & Hong Li & Zhichao Fang & Yang Liu, 2019. "A Mixed Finite Volume Element Method for Time-Fractional Reaction-Diffusion Equations on Triangular Grids," Mathematics, MDPI, vol. 7(7), pages 1-18, July.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:7:p:600-:d:246062
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/7/7/600/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/7/7/600/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Zhang, Tie & Li, Zheng, 2015. "An analysis of finite volume element method for solving the Signorini problem," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 830-841.
    2. Metzler, Ralf & Klafter, Joseph, 2000. "Boundary value problems for fractional diffusion equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 278(1), pages 107-125.
    3. Feng, L.B. & Zhuang, P. & Liu, F. & Turner, I., 2015. "Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 52-65.
    4. Zhao, Yanmin & Bu, Weiping & Huang, Jianfei & Liu, Da-Yan & Tang, Yifa, 2015. "Finite element method for two-dimensional space-fractional advection–dispersion equations," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 553-565.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Jie Zhao & Zhichao Fang & Hong Li & Yang Liu, 2020. "A Crank–Nicolson Finite Volume Element Method for Time Fractional Sobolev Equations on Triangular Grids," Mathematics, MDPI, vol. 8(9), pages 1-17, September.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Zeid, Samaneh Soradi, 2019. "Approximation methods for solving fractional equations," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 171-193.
    2. Qu, Wei & Li, Zhi, 2021. "Fast direct solver for CN-ADI-FV scheme to two-dimensional Riesz space-fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 401(C).
    3. Yin, Baoli & Liu, Yang & Li, Hong, 2020. "A class of shifted high-order numerical methods for the fractional mobile/immobile transport equations," Applied Mathematics and Computation, Elsevier, vol. 368(C).
    4. Marseguerra, Marzio & Zoia, Andrea, 2008. "Pre-asymptotic corrections to fractional diffusion equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(12), pages 2668-2674.
    5. Khan, Sharon & Reynolds, Andy M., 2005. "Derivation of a Fokker–Planck equation for generalized Langevin dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 350(2), pages 183-188.
    6. Nyamoradi, Nemat & Rodríguez-López, Rosana, 2015. "On boundary value problems for impulsive fractional differential equations," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 874-892.
    7. D’Ovidio, Mirko, 2012. "From Sturm–Liouville problems to fractional and anomalous diffusions," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3513-3544.
    8. Wei, T. & Li, Y.S., 2018. "Identifying a diffusion coefficient in a time-fractional diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 151(C), pages 77-95.
    9. Saffarian, Marziyeh & Mohebbi, Akbar, 2022. "Finite difference/spectral element method for one and two-dimensional Riesz space fractional advection–dispersion equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 348-370.
    10. Hosseininia, M. & Heydari, M.H., 2019. "Meshfree moving least squares method for nonlinear variable-order time fractional 2D telegraph equation involving Mittag–Leffler non-singular kernel," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 389-399.
    11. Zhang, Jingyuan, 2018. "A stable explicitly solvable numerical method for the Riesz fractional advection–dispersion equations," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 209-227.
    12. Wang, Shaowei & Zhao, Moli & Li, Xicheng, 2011. "Radial anomalous diffusion in an annulus," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(20), pages 3397-3403.
    13. Xian, Jun & Yan, Xiong-bin & Wei, Ting, 2020. "Simultaneous identification of three parameters in a time-fractional diffusion-wave equation by a part of boundary Cauchy data," Applied Mathematics and Computation, Elsevier, vol. 384(C).
    14. Zhang, Shougui, 2018. "Two projection methods for the solution of Signorini problems," Applied Mathematics and Computation, Elsevier, vol. 326(C), pages 75-86.
    15. Mophou, G. & Tao, S. & Joseph, C., 2015. "Initial value/boundary value problem for composite fractional relaxation equation," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 134-144.
    16. Zieniuk, Eugeniusz, 2017. "Approximation of the derivatives of solutions in a normalized domain for 2D solids using the PIES methodAuthor-Name: Bołtuć, Agnieszka," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 138-155.
    17. Guo, Gang & Li, Kun & Wang, Yuhui, 2015. "Exact solutions of a modified fractional diffusion equation in the finite and semi-infinite domains," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 417(C), pages 193-201.
    18. Marseguerra, M. & Zoia, A., 2008. "Monte Carlo evaluation of FADE approach to anomalous kinetics," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 77(4), pages 345-357.
    19. Al-Mdallal, Qasem M., 2009. "An efficient method for solving fractional Sturm–Liouville problems," Chaos, Solitons & Fractals, Elsevier, vol. 40(1), pages 183-189.
    20. Kashfi Sadabad, Mahnaz & Jodayree Akbarfam, Aliasghar, 2021. "An efficient numerical method for estimating eigenvalues and eigenfunctions of fractional Sturm–Liouville problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 185(C), pages 547-569.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:7:y:2019:i:7:p:600-:d:246062. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.